QUANTUM MECHANICS AND MOLECULAR STRUCTURE

6 QUANTUM MECHANICS AND MOLECULAR STRUCTURE 6.1 Quantum Picture of the Chemical Bond 6.2 Exact Molecular Orbital for the Simplest Molecule: H + 2 6.3 Molecular Orbital Theory and the Linear Combination of Atomic Orbitals Approximation for H + 2 6.4 Homonuclear Diatomic Molecules: First-Period Atoms 6.5 Homonuclear Diatomic Molecules: Second-Period Atoms 6.6 Heteronuclear Diatomic Molecules 6.7 Summary Comments for the LCAO Method and Diatomic Molecules 235 Potential energy diagram for the decomposition of the methyl methoxy radical

6.1 QUANTUM PICTURE OF THE CHEMICAL BOND 237 Specific example of H 2, (in section 3.7) V = - + + + + + V = V en +V ee +V nn Using effective potential energy function, V eff - At large R AB,V eff 0, and the atoms do not interact. - As R AB decreases, V eff must become negative because of attraction. - At very small R AB,V eff must become positive and large as V eff. 237 E 0 : zero-point energy for the molecule by the uncertainty principle Dissociation energy: D o or D e

238 effective potential energy function Born-Oppenheimer Approximation: Slow Nuclei, Fast Electrons 239 - Nuclei are much more massive than the electrons, the nuclei in the molecules will move much more slowly than the electrons. decoupling of the motions of the nuclei and the electrons (A) Consider the nuclei to be fixed at a specific set of positions. Then solve Schrödinger s equation for the electrons moving around and obtain the energy levels and wave functions. Next, move the nuclei a bit, and repeat the calculation. Continue this procedure in steps. Getting each electronic energy () to the nuclear coordinates RAB : the proper set of quantum numbers

239 (B) Consider the function () to be the attractive portion. Add the repulsive interaction to obtain the effective potential energy curve. () = () + 240 - Visualizing a group of electrons moving rapidly around the sluggish nuclei, to establish a dynamic distribution of electron density. rapid movement of the electrons effective potential energy functions

Mechanism of Covalent Bond Formation 241 The average value of (), () = + - When R AB, independent H atoms - 241 - As the two atoms approach one another, increases; the Coulomb attraction reduces between each electron and the proton to which it was originally bound. decreases; each electron becomes less confined. (The energy of the particle in a box decrease as the size of the box increases) decreasing the () -

241 - As bond formation continues, decreases; the simultaneous Coulomb attraction of each electron to two protons decreases the average potential energy. increases; confinement of the electron to the smaller internuclear region increases its kinetic energy. decreasing the () - 241 - The minimum in the (), and the equilibrium bond length of the molecule, is determined by the competition between the increasing average kinetic energy and the decreasing average potential energy at small values of internuclear separation. -

For the ionic bond, potential energy alone is essential. (section 3.8) 241 V(R 12 ) = Ae -R 12 - B ()() repulsion attraction + E For the covalent bond, the charge distribution and the kinetic energy of the electrons are also important. 6.2 EXACT MOLECULAR ORBITALS FOR THE SIMPLEST MOLECULE: H 2 + 242 H 2+ ion: a single electron bound to two protons bond length 1.06 Å; bond dissociation energy 2.79 ev = 269 kj mol -1 V = - + + = V en +V nn - For a fixed value of R AB, the position of the electron:,,,, - The potential energy has cylindrical (ellipsoidal) symmetry around the R AB axis.

242 By the Born-Oppenheimer approximation, -R AB : holding at the equilibrium bond length of 1.06 Å,, ; : the electronic wave function for the electrons Omitting due to the same potential energy for all values of The solution of the Schrödinger equation: - smooth, single-valued, and finite in all regions of space to define a probability density function of its square - physical boundary conditions 0 as and as Solutions exists when the total energy and angular momentum are quantized. Electronic Wave Functions for H 2 + 243 - isosurface comprising the wave function with 0.1 of its maximum value. - red: + amplitude; blue: - amplitude - molecular orbital: each of exact one-electron wave functions

243 - Four labels summarize the energy and the shape of each wave function. 1) integer: an index tracking the relative energy of the wave functions of each symmetry type. i.e.) 1 g : the first (the lowest energy) of the g wave functions 2) Greek letter: how the amplitude of the wave function is distributed around the internuclear axis. - : the amplitude with cylindrical symmetry around the axis - : the amplitude with a nodal plane that contains the internuclear axis 244 3) g or u: how the wave function changes as we invert our point of observation through the center of the molecule - the wave function at (x, y, z) and (-x, -y, -z) - g : symmetric, the same at these points gerade u : antisymmetric, the opposite at these points ungerade 4) * : how the wave function changes when the point of observation is reflected through a plane perpendicular to the internuclear axis - no symbol : no changing sign upon reflection * : changing sign upon reflection

Nature of the Chemical Bond in H 2 + 246 antibonding MO bonding MO Summary of the Quantum Picture of Chemical Bonding 247 1. The Born-Oppenheimer approximation: fixing the nuclei position 2. Molecular orbital: one-electron wave function, its square describes the distribution of electron density 3. Bonding MO: increased e density between the nuclei decreased effective potential energy 4. Antibonding MO: a node on the internuclear axis high effective potential energy 5. orbital: cylindrical symmetry; cross-sections perpendicular to the internuclear axis are discs. 6. orbital: a nodal plane containing the internuclear axis

6.3 MOLECULAR ORBITAL THEORY AND THE LINEAR COMBINATION OF ATOMIC ORBITALS APPROXIMATION FOR H 2 + 248 LCAO method: selecting sums and differences (linear combinations) of atomic orbitals to generate the best approximation to each type of molecular orbital - The general form for H 2 + = ± : generic wave function : wave function of atomic orbitals The two nuclei are identical, then =± 248 MOs of the bonding = + = The distribution of electron probability density = + + = +

249 250

Energy of H 2+ in the LCAO Approximation 251 g1s, R AB = 1.32 Å, D = 1.76 ev g, R AB = 1.06 Å, D 0 = 2.79 ev Correlation diagram: the energy-level diagram within the LCAO 251

6.4 HOMONUCLEAR DIATOMIC MOLECULES: FIRST-PERIOD ATOMS For He 2+ and He 2, 252 A B A B g1sc ghe1s He1s C g1s 1s * A B A B u1sc uhe1s He1s C g1s 1s Fig. 6.11. Correlation diagram for H 2. Fig. 6.12. Correlation diagram for He 2 H 2 : Stabilization of bonding MO 252 by 2( E) for H 2 compared to the noninteracting system. He 2+ : Stabilization of bonding MO by 2( E) compensated by destabilization of antibonding MO by +E. Net stabilization energy = E

Bond order 253 Bond order = (1/2) x (number of electrons in bonding MOs number of electrons in antibonding MOs) 6.5 HOMONUCLEAR DIATOMIC MOLECULES: SECOND-PERIOD ATOMS 253 LCAO-MO approximation Combination of 2s AOs to form g2s and u2s * MOs g2s C s s A B g[2 2 ] C [2s 2 s ] * A B u2s u Combination of 2p z AOs to form g p 2 z u p * and MOs 2 z A B g2 pcg[2pz 2 pz] z C [2p 2 p ] * A B u2 p u z z z

256 g p * u p Fig. 6.14. Formation of (a) 2 z bonding and (b) 2 z antibonding MOs from 2p z orbitals on atoms A and B. u 2 py Doubly degenerate & MO s 2 p x 257 A B u2 p Cu[2px 2 px] x C [2p 2 p ] * A B g2 p g x x x A B u2 p Cu[2py 2 py] y C [2p 2 p ] * A B g2 p g y y y

258 Determination of energy ordering 1. Average energy of bonding-antibonding pair of MOs similar to that of original AO s 2. Energy difference between a bonding-antibonding pair becomes large as the overlap of AO s increases 258 Fig. 6.16. Energy levels for the homonuclear diatomics Li 2 through F 2.

259 Fig. 6.17. Correlation diagrams for second-period diatomic molecules, N 2 & F 2. 259

259 260 Reversed ordering of energy for Li 2 ~N 2 (or ) g2p u2p u2p z x y ~ Due to large electron-electron spatial repulsions between g p electrons in and MO s. 2 z * u 2 s Normal ordering of energy for O 2, F 2, Ne 2 (or ) ~ As Z increases, the repulsion decreases since electrons * in and MO s are drawn more strongly toward the nucleus. g2p u2p u2p z x y g2s u2s

260 Fig. 6.18. (a) Paramagnetic liquid oxygen, O 2, and (b) diamagnetic liquid nitrogen, N 2, pours straight between the poles of a magnet. 261 Fig. 6.19. Trends in several properties with the number of valence electrons in General the second-row Chemistry diatomic I molecules.

6.6 HETERONUCLEAR DIATOMIC MOLECULES 262 For the 2s orbitals, 2s = C A 2s A + C B 2s B 2s * = C A 2s A C B 2s B In the homonuclear case, C A = C B ; C A = C B If B is more electronegative than A, C B >C A for bonding s MO, C A > C B for the higher energy s* MO, closely resembling a 2s A AO 263 Fig. 6.20. Correlation diagram for heteronuclear diatomic General Chemistry molecules, I BO. (O is more electronegative.)

265, Fig. 6.22. Correlation diagram for HF. The 2s, 2p x, and 2p y atomic orbitals of General F do not Chemistry mix with the I 1s atomic orbital of H, and therefore remain nonbonding. 264 * Fig. 6.21. Overlap of atomic orbitals in HF.

6.7 SUMMARY COMMENTS FOR THE LCAO METHOD AND DIATOMIC MOLECULES 265 The qualitative LCAO method easily identifies the sequence of energy levels for a molecule, such as trends in bond length and bond energy, but does not give their specific values. The qualitative energy level diagram is very useful for interpreting experiments that involve adsorption and emission of energy such as spectroscopy, ionization by electron removal, and electron attachment. 6 QUANTUM MECHANICS AND MOLECULAR STRUCTURE 6.8 Valence Bond Theory and the Electron Pair Bond 6.9 Orbital Hybridization for Polyatomic Molecules 6.10 Predicting Molecular Structures and Shapes 6.11 Using the LCAO and Valence Bond Methods Together 6.12 Summary and Comparison of the LCAO and Valence Bond Methods

6.8 VALENCE BOND THEORY AND THE ELECTRON PAIR BOND 268 Explains the Lewis electron pair model VB wave function for the bond is a product of two one-electron AOs Easily describes structure and geometry of bonds in polyatomic molecules Nobel Prizes Chemistry ( 54) The Nature of Chemical Bonding Peace ( 62) Walther Heitler Fritz London (DE, 1904-1981) (DE, 1900-1954) John C. Slater (US, 1900-1976) Linus Pauling (US,1901-1994) Single Bonds 268 - At very large values of R AB,, ; = independent atoms - As the atoms begin to interact strongly,, ; = + indistinguishable =±

VB wave function for the single bond in a H 2 molecule g C1 1 s (1)1 s (2) 1 s (2)1 s (1) Bonding Wavefunction MO el A B A B u C1 1 s (1)1 s (2) 1 s (2)1 s (1) Antibonding Wavefunction MO el A B A B 269 Fig. 6.24. (a) The electron density g for el g and u for el u in the simple VB model for H 2. (b) Three-dimensional isosurface of the electron density General for the Chemistry el g wave function I in the H 2 bond. For F 2 bond, 270 = 2 12 2 +2 (2)2 (1) For HF bond, = 2 11 2 + 2 21 1 H F

Multiple Bonds 271 N: (1s) 2 (2s) 2 (2p x ) 1 (2p y ) 1 (2p z ) 1 = 2 12 2 +2 (2)2 (1) 1,2 = 2 12 2 + 2 22 1 1,2 = 2 12 2 + 2 22 1 Polyatomic Molecules 273? all C-H bonds equivalent Electron promotion: electron relocated to a higher-energy orbital promotion hybridization

6.9 ORBITAL HYBRIDIZATION FOR POLYATOMIC MOLECULES 273 sp-hybridization BeH 2 Promotion: Be: (1s) 2 (2s) 2 Be: (1s) 2 (2s) 1 (2p z ) 1 Two equivalent sp hybrid orbitals: 1 1 and 1() r 2s 2p z 2() r 2s2p z 2 2 New electronic configuration, Be: (1s) 2 ( 1 ) 1 ( 2 ) 1 Wave functions for the two bonding pairs of electrons: bond H H 1 (1, 2) c 1(1)1 s (2) 1(2)1 s (1) bond H H 2 (3, 4) c 2(3)1 s (4) 2(4)1 s (3) A pair of bonds at an angle 180 o apart linear molecule 275 Fig. 6.28. Formation, shapes, and bonding of the sp hybrid orbitals in the BeH 2 molecule. (a) The 2s and 2p z orbitals of the Be atom. (b) The two sp hybrid orbitals formed from the 2s and 2p z orbitals on the Be atom. (c) The two bonds that form from the overlap of the sp hybrid orbitals with the H1s orbitals, making tow single bonds in the BeH 2 molecule. (d) Electron density in the two bonds.

275 sp 2 -hybridization BH 3 Promotion: B: (1s) 2 (2s) 2 (2p x ) 1 B: (1s) 2 (2s) 1 (2p x ) 1 (2p y ) 1 Three equivalent sp 2 hybrid orbitals: = +2 / 2 = + 3/2 / 2 1/2 / 2 = 3/2 / 2 1/2 / 2 New electronic configuration, B: (1s) 2 ( 1 ) 1 ( 2 ) 1 ( 3 ) 1 Three bonds at an angle 120 o in a plane trigonal planar 276 General eral Chemistry I

276 sp 3 -hybridization CH 4 Promotion: C: (1s) 2 (2s) 2 (2p x ) 1 (2p y ) 1 C: (1s) 2 (2s) 1 (2p x ) 1 (2p y ) 1 (2p z ) 1 Four equivalent sp 3 hybrid orbitals: 1 1() r 2s2px 2py 2pz 2 1 2() r 2s2px 2py 2pz 2 1 3() r 2s2px 2py 2pz 2 1 4() r 2s2px 2py 2pz 2 New electronic configuration, C: (1s) 2 ( 1 ) 1 ( 2 ) 1 ( 3 ) 1 ( 4 ) 1 Four bonds at an angle 109.5 o generating tetrahedral geometry 277 Fig. 6.30. Shapes and relative orientations of the four sp 3 hybrid orbitals in CH 4 pointing at the corners of a tetrahedron with the C atom at its center.

Summary of Hybridization Results 277 Hybridization and Lone Pairs 278 NH 3 : sp 3 hybrid orbitals trigonal pyramid with three equivalent bonds H 2 O: sp 3 hybrid orbitals two lone pairs the bent or angular structure

Multiple Bonds in Organic Compounds 278 Promotion, C: (1s) 2 (2s) 2 (2p x ) 1 (2p y ) 1 C: (1s) 2 (2s) 1 (2p x ) 1 (2p y ) 1 (2p z ) 1 Formation of three sp 2 hybrid orbitals from the 2s and two 2p orbitals New electronic configuration, C: (1s) 2 ( 1 ) 1 ( 2 ) 1 ( 3 ) 1 (2p z ) 1 Five bonds: C 1 1 -H1s, C 1 2 -H1s, C 2 1 -H1s, C 2 1 -H1s, C 1 3 -C 2 3 One bond: C 1 2p z -C 2 2p z One double bond (C=C): (C 1 3 -C 2 3 ) + (C 1 2p z -C 2 2p z ) 279 Fig. 6.33. Formation of bonds in ethylene. General Chemistry Fig. 6.34. I Formation of a bond in ethylene.

280 Linear, triple bond, H-C-C bond angles of 180 o Promotion, C: (1s) 2 (2s) 2 (2p x ) 1 (2p y ) 1 C: (1s) 2 (2s) 1 (2p x ) 1 (2p y ) 1 (2p z ) 1 Formation of two sp hybrid orbitals from the 2s and the 2p z orbitals New electronic configuration, C: (1s) 2 ( 1 ) 1 ( 2 ) 1 (2p x ) 1 (2p y ) 1 Three bonds: C 1 1 -H1s, C 2 1 -H1s, C 1 2 -C 2 2 Two bonds: C 1 2p x -C 2 2p x, C 1 2p y -C 2 2p y One triple bond : (C 1 2 -C 2 2 ) + (C 1 2p x -C 2 2p x, C 1 2p y -C 2 2p y ) 280 Fig. 6.35. Formation of bonds in acetylene. General Chemistry Fig. 6.36. I Formation of two bonds in acetylene.

281 6.10 PREDICTING MOLECULAR STRUCTURES AND SHAPE Description of the structure and shape of a molecule (1) Determine the empirical formula. (2) Determine the molecular formula. (3) Determine the structural formula from a Lewis diagram. (4) Determine the molecular shape from experiments. (5) Identify the hybridization scheme that best explains the shape predicted by VSEPR. EXAMPLE 6.4 Hydrizine: Elemental analysis shows its mass per cent composition to be 87.419% nitrogen and 12.581% hydrogen. The density of hydrazine at 1 atm and 25 o C is 1.31 gl -1. Determine the molecular formula for hydrazine. Predict the structure of hydrazine. What is the hybridization of the N atoms? (1) Elemental analysis: N(87.419%), H (12.581%) (2) Empirical formula: NH 2 (Molar mass) emp (3) Molar mass calculate from the ideal gas law (with known ) (4) Molar mass / (Molar mass) emp = 2, Molecular formula: N 2 H 4 (5) Lewis diagram, steric number (4) sp 3 hybrid orbitals 282

283 The electrostatic potential energy map 284 Fig. 6.38. (a) Space-filling models. (b) 0.002 e/(a 0 ) 3 electron density surfaces, and (c) electrostatic potential energy surfaces.

6.11 USING THE LCAO AND VALENCE BOND METHODS TOGETHER 287 Linear Triatomic Molecules (CO 2, N 2 O, NO 2+ ) Central atom sp hybridization from s and p z orbitals Two sp orbitals form bonds with outer atoms Remaining p x and p y orbitals forms bond bonds p z orbital (outer) with sp hybrid (central) bonds linear combinations of the p x (or p y ) orbitals bonding, antibonding, and nonbonding 287

288 288 Nonlinear Triatomic Molecules (NO 2, O 3, NF 2, NO 2- ) Central atom sp 2 hybridization from s, p x, and p y orbitals One of sp 2 orbitals holds a lone pair Two of sp 2 orbitals form bonds with outer atoms Remaining p z orbital forms bond Outer atom p orbital pointing toward the central atom forms a bond p z orbital forms a bond with p z orbitals of other atoms Remaining p orbital and s orbital nonbonding

289 Fig. 6.42. (a) Ball-and-stick model (bottom) and molecular orbitals for bent triatomic molecule, NO 2, with three sp 2 hybrid orbitals on the central N atom that would lie in the plane of the molecule. (b) Correlation diagram for the orbitals. There is only one of each of orbitals (, nb, *). 6.12 SUMMARY AND COMPARISON OF THE LCAO AND VALENCE BOND METHODS 290 LCAO method for H 2 forming a bond: A B A B g1scg 1s1 s 1s 1s Molecular electronic wave function in the LCAO approximation: el A B A B MO g1s(1) g1s(2) 1 s (1) 1 s (1) 1 s (2) 1 s (2) el A B A B A A B B s s s s s s s s MO 1 (1)1 (2) 1 (2)1 (1) 1 (1)1 (2) 1 (1)1 (2)

291 Molecular electronic wave function in the VB model: ) 1) el el A B A B VB( r1a, r2b ) c1 A ( r1a ) B( r2b ) c2 A ( r2a ) B( r1b VB 1 s (1)1 s (2) 1 s (2)1 s ( Comparison of MO and VB theories: = 1 11 2 +1 (2)1 (1) + 1 11 2 +1 (1)1 (2) purely covalent structure H H mixture of ionic states, H A- H B+ and H A+ H B - Improved VB wavefunction: improved VB ionic 293